Elliptically-shaped tool

ABSTRACT

Producing a tool for creating a mandrel for forming elliptically-shaped lens elements of a lenticular lens array comprises the steps of providing a base member having a radius b corresponding to a minor axis of an elliptical shape, the elliptical shape corresponding to a desired elliptical shape for each lens element of the lenticular lens array, and cutting the base member along a plane that forms an angle k with the minor axis of the desired elliptical shape. The elliptical shape comprises a major axis perpendicular to the minor axis, a vertex of the desired elliptical shape lies at a point a along the major axis, and the angle k is given by the formula cosine(k)=b/a.

PRIORITY AND RELATED APPLICATIONS

This application is a continuation of U.S. Non-provisional PatentApplication No. 10/167,020, entitled “Lenticular Lens Array and Tool forMaking a Lenticular Lens Array.” filed Jun. 10, 2002, which claims thebenefit of priority to U.S. Provisional Patent Application Serial No.60/297,148, entitled “Lenticular Lens Array Optimization for PrintedDisplay,” filed Jun. 8, 2001. The complete disclosure of each of theabove-identified priority applications is fully incorporated herein byreference.

FIELD OF THE INVENTION

The present invention relates generally to a lenticular lens array forproducing visual effects from interdigitated or interlaced images. Moreparticularly, the present invention relates to a lenticular lens arraywhere a cross section of each lens element on the array comprises anelliptical shape. The present invention also relates to a tool and amethod for creating such a lenticular lens array.

BACKGROUND OF THE INVENTION

A lenticular lens can create visual animated effects for interdigitatedor interlaced (hereinafter “interlaced”) printed images. The images canbe printed using non-impact printing, known as masterless printing, orby conventional printing processes, known as master printing. Typically,a lenticular lens application comprises two major components: anextruded, cast, or embossed plastic lenticular lens and the interlacedprinted image. The front of the lenticular lens comprises a plurality oflenticules arranged in a regular array, having cylindrical lens elementsrunning parallel to one another. The back of the lenticular lens is flatand smooth. The interlaced images are printed on the flat, smoothbackside of the lenticular lens. Exemplary methods for printing theimages include conventional printing methods such as screen,letterpress, flexographic, offset lithography, and gravure; andnon-impact printing methods such as electro-photography, iconography,magnetography, ink jet, thermography, and photographic. Any of the aboveprinting technologies can be used in either sheet-fed or roll web-fedforms.

The interlaced images are viewed individually, depending on the anglethrough which a viewer observes the images through the lenticular lenselements. At a first viewing angle, a first image appears through thelenticular lens elements. As the lenticular lens is rotated, the firstimage disappears and another; image appears through the lenticular lenselements. Viewing the images through the lenticular lens elements cancreate the illusion of motion, depth, and other visual effects. Alenticular lens can create those illusions through different visualeffects. For example, the visual effects can comprise three-dimensions(3-D), animation or motion, flip, morph, zoom, or combinations thereof.

For a 3-D effect, multiple layers of different visual elements areinterlaced together to create the illusion of 3-D, distance, and depth.For example, background objects are pictured with foreground objectsthat appear to protrude when viewed through a straight forward,non-angled view. For an animation or motion effect, a series ofsequential photos can create the illusion of animated images. A viewerobserves the series of photos as the viewing angle of the lens changes.Animation is effective in showing mechanical movement, body movement, orproducts in use.

For a flip visual effect, two or more images flip back and forth as theviewing angle changes. The flip effect can show before-and-after andcause-and-effect scenarios. It also can show bilingual messages, such asflipping from English to Spanish. For a morph visual effect, two or moreunrelated images gradually transform or morph into one another as theviewing angle of the lenticular lens changes. Finally, for a zoomeffect, an object moves from the background into the foreground as theviewing angle of the lenticular lens changes. The object also may travelfrom side to side, but usually works better in a top to bottom format.

FIG. 1 illustrates a partial cross section of a conventional lenticularlens array 100. The array 100 comprises lenticules 102, 104, 106. Eachlenticule 102, 104, 106 comprises a cylindrical lens element 102 a, 104a, 106 a, respectively. Each lens element 102 a, 104 a, 106 a operatesto focus light on a back surface 107 of the array 100. In operation ofthe conventional array 100, multiple images can be printed on the rearsurface 107. An observer can singularly view the images through the lenselements 102 a, 104 a, 106 a by rotating the array 100.

Specific characteristics of each lenticule 102, 104, 106 will bedescribed with reference to exemplary lenticule 104. Each lens element102 a, 104 a, 106 a has a circular cross section of radius R. Thecircular cross section corresponds to a desired circular shape 108having the radius R. The lens element 104 a comprises a portion of thecircular shape 108. Lenticule 104 also has a distance t from a vertex ofthe lens element 104 a to the rear surface 107 of the array 100. Thelens element 104 a has a lens junction depth d where it joins adjacentlens elements 102 a, 106 a. Finally, the material forming the lens array100 determines a refractive index N of the array 100.

The relationship between the distance t, the radius R, and therefractive index N is given by the following equation: $\begin{matrix}{t = \frac{RN}{N - 1}} & (1)\end{matrix}$

As shown in equation (1), the thickness t and radius R are a function ofthe refractive index N, which is a function of wavelength of light.Accordingly, the lenticular lens elements can be optimized for aparticular wavelength based on the wavelength that provides the bestoverall performance for the desired application.

Regularity of the array 100 can be defined by the separation or distanceS between the vertex of adjacent lens elements. For the conventionalcylindrical lenticular lens array 100, the maximum separation betweenthe vertex of each lens element 102 a, 104 a, 106 a is given by thefollowing equation:

S_(max=)2R  (2)

A pitch P of the lenticules can be defined as a number of lenticules perunit length (lpu). For example, the unit length can comprise an inch ora millimeter. For the conventional cylindrical lenticular lens array100, the minimum pitch is given by the following equation:$\begin{matrix}{P_{\min} = {\frac{1}{2R}\lbrack{lpu}\rbrack}} & (3)\end{matrix}$

FIG. 2 illustrates a light ray trace illustrating several problemsassociated with a conventional lenticular lens array 100. In general,the array 100 operates by passing light from the rear surface 107through the lens elements 102 a, 104 a, 106 a to an observer.Reciprocity allows viewing the light path in reverse as illustrated inFIG. 2. Ideally, on-axis light L₁ passes through lens element 104 a andis focused to a common point 202 on the rear surface 107 of the array100. However, the circular cross-section of the lens element 104 aproduces a projected image having spherical aberration. For example, thelight L₁ is projected over a large area 204 on the rear surface 107. Thelarge projection area limits resolution and the number of interlacedimages that can be viewed on the rear surface 107.

Additionally, off-axis light L₂ passes through the lens element 104 aand is focused upon the rear surface 107 near point 203. However, thecircular cross-section of lens element 104 a produces coma and anastigmatic aberration 208. Finally, FIG. 2 illustrates that the depth dof the lens surface can approach the radius of the circularcross-section at the junction of adjacent lenses. Accordingly, portionsof the light L₂ are blocked by lens 106 a and may be redirected to thewrong location 206.

FIG. 3 illustrates a light beam projection illustrating another problemassociated with the conventional lenticular lens array 100. FIG. 3illustrates light beams projected to an observer from different printedareas of the conventional lenticular lens array 100. As shown, the lightbeams in the central area 302 are not reasonably matched over thecircular angle of the lens.

Furthermore, conventional lenticular sheet-fed printing has been used tocreate promotional printed advertising pieces printed on a lenticularlens array. For example, the advertising pieces include limited volumesof thicker gauge lenticular material designs such as buttons, signage,hang tags for clothing, point-of-purchase displays, postcards, greetingcards, telephone cards, trading cards, credit cards, and the like. Thosethicker gauge lenticular printed products are printed on cylindricallenticular material having a standard thickness. For example, standardthicknesses include 0.012 mil, 0.014 mil, 0.016 mil, 0.018 mil, and upto 0.0900 mil. Printed quality on those thicker lenses are generallyacceptable because the lenticule pitch is more course (fewer lenticules)and the printing process can place more printed image pixels within thelenticule band range. Additionally, lenticular materials at the thickerranges tend to be more optically forgiving then thinner gauges.

Recently, lenticular extruders, lenticular casting/embossers, and printmanufacturers have experimented with decreasing the overall lenticularmaterial thickness using the common cylindrical lens elements discussedabove. However, as the thickness of the lenticular lens array decreases,the print quality suffers significant aberration. As the thicknessdecreases, lenticule pitch must increase to provide more lenticules perunit length, thereby reducing the separation between lenticules. Thatthinner configuration does not allow using as many printed pixel imageswhen compared to the thicker lenticular material designs. Accordingly,the quality of the printed visual effects is degraded with the thinnermaterial.

Another problem with thicker lenticular materials is that the thickermaterials cannot be used for the majority of the consumer packagingindustry. That problem arises because thicker materials of 0.012 mil andthicker cannot be applied nor handled properly to cylindrical ortruncated package shapes without de-laminating off the package due toplastic memory pull. Even when a strong adhesive is used to bond thethick lenticular piece to the packaged unit, problems with de-laminationstill occur over time due to the continual pull of the plastic material,as the plastic memory pulls the material to its natural, straightproduced shape.

Thicker lenticular materials also experience problems during the labelapplication process. Automated printed label blow-down or wipe-downpackaging labeling equipment cannot apply the thicker lenticularmaterials, because of the plastic memory issues discussed above. Theplastic memory causes the thicker lenticular die cut labels to rise offthe lenticular label rolls before the application process.

Therefore, a need in the art exists for a lenticular lens array that canprovide a more focused or resolved image by mitigating the sphericalaberration associated with conventional arrays. A need in the art alsoexists for a tool and a method for making such a lenticular lens array.Furthermore, a need exists in the art for a lenticular lens array havinga lenticular lens element shaped to mitigate the spherical aberrationassociated with conventional lenticular lens elements. A need alsoexists for a lenticular lens array having a thin structure to mitigateplastic memory issues associated with thicker, conventional arrays.

SUMMARY OF THE INVENTION

The present invention can provide a lenticular lens array that canoptimize printed display quality of animated/three-dimensional imagesfor mass production.

The present invention can provide a lenticular lens array that canmitigate the spherical aberration typically produced by a conventionalarray. For example, the present invention can provide a lenticular lensarray that can produce a substantially focused axial image and canimprove the off-axis image. Additionally, the present to invention canprovide a lenticular lens array having a reduced lens junction depth,which can mitigate off-axis light blocking by adjacent lenses.

The lenticular lens array according to the present invention cancomprise a plurality of lenticules disposed adjacent to each other. Eachlenticule can comprise a lenticular lens element on one side and asubstantially flat surface on an opposite side. Each lenticular lenselement can have a vertex and a cross section comprising a portion of anelliptical shape. Alternatively, the cross section can comprise anapproximated portion of an elliptical shape. The elliptical shape cancomprise a major axis disposed substantially perpendicular to thesubstantially flat surface of each respective lenticular lens element.The vertex of each respective lenticular lens element can liesubstantially along the major axis of the elliptical shape.

These and other aspects, objects, and features of the present inventionwill become apparent from the following detailed description of theexemplary embodiments, read in conjunction with, and reference to, theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a partial cross section of a conventional lenticularlens array.

FIG. 2 illustrates a light ray trace illustrating problems associatedwith a conventional lenticular lens array.

FIG. 3 illustrates a light beam projection illustrating another problemassociated with the conventional lenticular lens array.

FIG. 4 illustrates a partial cross section of a lenticular lens arrayaccording to an exemplary embodiment of the present invention.

FIG. 5 illustrates a light ray trace illustrating opticalcharacteristics of a lenticular lens array according to an exemplaryembodiment of the present invention.

FIG. 6 illustrates a light beam projection illustrating additionaloptical characteristics of the lenticular lens array according to anexemplary embodiment of the present invention.

FIG. 7 illustrates a partial cross section of a lenticular lens arrayaccording to an alternative exemplary embodiment of the presentinvention.

FIG. 8A illustrates a cross-section of a cylindrical rod for producing atool for forming elliptically-shaped lens elements according to anexemplary embodiment of the present invention.

FIG. 8B illustrates a front view of the tool for formingelliptically-shaped lens elements according to an exemplary embodimentof the present invention.

FIG. 9 illustrates a cross section of a pseudo elliptical lenticule forapproximating an elliptically shaped lens element of a lenticular lensarray according to an exemplary embodiment of the present invention.

FIG. 10 illustrates a pseudo elliptical tool for creating a pseudoelliptical lens element according to an exemplary embodiment of thepresent invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

The present invention can reduce spherical aberration associated withconventional lenticular lens arrays by providing a lenticular lens arrayhaving an elliptical cross-sectional shape. The ellipticalcross-sectional shape can provide sharp focusing of on-axis light andcan increase the clarity of off-axis light. The characteristic shape ofthe elliptical cross section can be determined based on a particularapplication. Many parameters can influence the elliptical shape. Forexample, the parameters include a refractive index N of the arraymaterial, a thickness t from the vertex of each lens element to a rearsurface of the array, a lens junction depth d where adjacent lensesjoin, and other parameters. A pseudo elliptical lens element also canprovide a lenticular lens array having reduced spherical aberration.

FIG. 4 illustrates a partial cross section of a lenticular lens array400 according to an exemplary embodiment of the present invention. Thearray 400 comprises lenticules 402, 404, 406. Each lenticule 402, 404,406 comprises an elliptically-shaped lens element 402 a, 404 a, 406 a,respectively. Each lens element 402 a, 404 a, 406 a operates to focuslight on a back surface 407 of the array 400. In operation of the array400, multiple images can be printed on the rear surface 407 of the array400. An observer can singularly view the images through the lenselements 402 a, 404 a, 406 a by rotating the array 400.

Specific characteristics of each lenticule 402, 404, 406 will bedescribed with reference to exemplary lenticule 404. Each lens element402 a, 404 a, 406 a has an elliptical cross section corresponding to aportion of a desired elliptical shape 408. The lens element 404 acomprises a portion of the elliptical shape 408. At a vertex of the lenselement 404 a, the elliptical shape 408 has a radius R. Lenticule 404also has a distance t from a vertex of the lens element 404 a to therear surface 407 of the array 400. The lens element 404 a has a lensjunction depth d where it joins adjacent lens elements 402 a, 406 a. Thematerial forming the lens array 400 determines a refractive index N ofthe array 400. The relationship between the distance t, the radius R,and the refractive index N is given by equation (1) discussed above.

The characteristics of the elliptical shape 408 will now be described.The elliptical shape 408 comprises an ellipse having a major axis 410and a minor axis 412. The ellipse crosses the major axis 410 at points±a and the minor axis 412 at points ±b. The major axis 410 and the minoraxis 412 cross at the origin o. The ellipse also comprises foci locatedat points ±c on the major axis 410. The junction point of adjacent lenselements 402 a, 404 a, 406 a crosses the elliptical shape 408 at adistance y from the major axis 410. The optical axis of the lenticule404, which is the major axis 410 of the elliptical shape 408, isperpendicular to the rear surface 407 of the array 400. The vertex ofthe lens element 404 a is positioned along the major axis 410 of theelliptical shape 408.

For the lenticular lens array 400, the maximum separation between thevertex of each lens element 402 a, 404 a, 406 a is given by thefollowing equation:

S _(max)=2b  (4)

The maximum separation between the vertex of each lens element 402 a,404 a, 406 a also is given by the following equation: $\begin{matrix}{S_{\max} = \frac{2{RN}}{\sqrt{N^{2} - 1}}} & (5)\end{matrix}$

The array 400 has a pitch defined by the number of lenticules per unitlength (lpu). For example, the unit length can comprise an inch or amillimeter. For the lenticular lens array 400, the minimum pitch isgiven by the following equation: $\begin{matrix}{P_{\min} = {\frac{1}{2b}\lbrack{lpu}\rbrack}} & (6)\end{matrix}$

Parameters for a particular application of the lenticular lens array 400can determine the characteristics of the elliptical shape 408. Thecharacteristics can be determined for each application. For example, thecharacteristics d, t, y, and R of the elliptical shape 408 can bedetermined from the refractive index of the material forming the array400 and standard geometric equations. For instance, the major axis 410can lie along an x-axis and the minor axis 412 can lie along a y-axis ofa rectangular coordinate system. Accordingly, the elliptical shape 408is given by the following equation:

y ²−2Rx+px ²=0  (7)

The constant p can be determined in terms of a conic constant; as shownin the following equation:

p=κ+1  (8)

The conic constant κ can define the elliptical shape of the lens 404 andcan be determined from the following equation: $\begin{matrix}{\kappa = {- \frac{1}{N^{2}}}} & (9)\end{matrix}$

The refractive index N is typically in the range of about 1.3 to about2.0, and more commonly in the range of about 1.5 to about 1.6, forplastics used in the printing industry. Accordingly, the conic constantκ for the bounding refractive index range covers from about −0.25 toabout −0.60. Accordingly, those conic constants indicate an ellipticalshape for the lens element 404 a, because a conic constant less thanzero and greater than minus one indicates an elliptical shape.

The eccentricity e of the elliptical shape 408 is given by the followingequations:

e={square root over (−κ)}, or  (10)

$\begin{matrix}{e = \frac{c}{a}} & (11)\end{matrix}$

Other standard geometric relationships for the elliptical shape 408include the following: $\begin{matrix}{{a = \frac{R}{P}},\quad {or}} & (12) \\{a = \frac{R}{\kappa + 1}} & (13)\end{matrix}$

 b ² =a ² −c ²  (14)

An example of determining particular characteristics for the ellipticalshape 408 will now be described. A desired material to form the array400 can be chosen. The desired material can have an associatedrefractive index N. Using the refractive index N, a conic constant κ forthe elliptical shape 408 can be determined using equation (9).Additionally, a lenticule thickness t can be chosen for the particularapplication. For example, the lens thickness t can be in the range ofabout 0.003 to about 0.100 inches. In an exemplary embodiment, thelenticule thickness t can be chosen in the range of about 0.007 to about0.011 inches. Alternatively, the lenticule thickness t can Using thestandard geometric equations, the points ±a and ±b that define theelliptical shape 408 can be determined. For example, the radius R can bedetermined using equation (1) and the conic constant κ from equation(9). Then, points ±a can be determined using equation (12) or (13).Next, the eccentricity e can be determined using equation (10). Points±c can be determined using equation (11). Points ±b can be determinedusing equation (14).

The distance y can be chosen based on the particular application for thearray 400. The distance y is one half the width of the lens element 404a. The width of the lens element 404 a can define a field of view forthe lens element 404 a on the rear surface 407. Accordingly, thedistance y can be chosen to provide a field of view wide enough for adesired number of interlaced images. After choosing the distance y, thex coordinate on the major axis 410 for the distance y can be determinedusing equation (7).

The particular characteristics of the elliptical shape 408 can bedetermined from many combinations of the parameters that define thosecharacteristics. Accordingly, the present invention encompassesdetermining elliptical characteristics based on a different set ofchosen or given initial parameters than those described above.

FIG. 5 illustrates a light ray trace illustrating opticalcharacteristics of the lenticular lens array 400 according to anexemplary embodiment of the present invention. The lenticular lens array400 can mitigate the spherical aberration typically produced by aconventional array. For example, the array 400 can provide asubstantially focused axial image and can improve the off-axis image. Asshown in FIG. 5, the on-axis light L₁ can pass through the lens element404 a of the array 400 and can be focused at point 502 on the rearsurface 407. As shown, the elliptically-shaped lens element 404 a canmitigate spherical aberration produced around the focal point 502. Byreducing the base spherical aberration, spherochromatism can also bereduced.

Additionally, the off-axis image produced from the off-axis light L₂ atpoint 503 is improved over the conventional lens, with coma 508 beingthe clear residual aberration. Also, the elliptically-shaped lenselement 404 a can reduce the lens junction depth d between adjacent lenselements. Accordingly, the array 400 can mitigate off-axis lightblocking by adjacent lenses, as shown in FIG. 4. For a given width 2y,radius R, and index of refraction N, ghosting can be reduced becauseoff-axis light blocking is reduced compared to a conventional circulararray having the same width, radius, and index of refractioncharacteristics.

FIG. 6 illustrates a light beam projection illustrating additionaloptical characteristics of the lenticular lens array 400 according to anexemplary embodiment of the present invention. FIG. 6 illustrates lightbeams projected to an observer from different printed areas of thelenticular lens array 400. As shown, the light beams in the central area602 are reasonably matched over the angle of the elliptically-shapedlens.

FIG. 7 illustrates a partial cross section of a lenticular lens array700 according to an alternative exemplary embodiment of the presentinvention. The array 700 can comprise the lenticular lens array 400coupled to a substrate 702. In the exemplary embodiment, the lenselements 402 a, 404 a, and 406 a can focus light on a rear surface 704of the substrate 702. The total distance T from each lens vertex to therear surface 704 of the substrate 702 can comprise the distance t₁ fromthe lens vertex to the rear surface 407 of the array 400 plus thedistance t₂ from the rear surface 407 of the array 400 to the rearsurface 704 of the substrate 702. In practice, the lenticular lens arrayis cast and has a thickness t₁ typically equal to about lens junctiondepth d or slightly greater than the lens junction depth d. Thecharacteristics of the elliptically-shaped lenses 402 a, 404 a, 406 acan be similar to those described above with reference to FIG. 4.

The array 400 and the substrate 702 can comprise different materials.Accordingly, the different materials can have different refractiveindexes. For example, the array 400 can comprise a material having arefractive index of N₁, and the substrate 702 can comprise a materialhaving a refractive index of N₂. The different refractive indices of thearray and substrate materials can introduce additional sphericalaberration. For example, for a single additional substrate of thicknesst₂ and refractive index N₂, the focal displacement is shifted withrespect to an array comprising a single material of refractive index N₁and having the same R. The shift in focal displacement can be eitherpositive or negative depending upon the relationship of the materials.To compensate for the different refractive indices, equation (1) can bemodified to the following equation to determine the radius R of eachlens element 402 a, 404 a, 406 a when the array 700 comprises two ormore different materials: $\begin{matrix}{R = {\left( {N_{1} - 1} \right)\left( {\frac{t_{1}}{N_{1}} + \frac{t_{2}}{N_{2}} + {\ldots \frac{t_{n}}{N_{n}}}} \right)}} & (15)\end{matrix}$

The value of the radius R results in the image from a distant sourcebeing formed upon the back surface of the substrate 702. As shown above,equation (15) can apply when the lenticular lens array comprises morethan one substrate.

The conic constant can be estimated from equation (9) and can beoptimized with an optical computer program to mitigate the additionallyinduced spherical aberration of the substrate(s).

In an alternative exemplary embodiment, the substrate 702 can be bondedto the array 400 through a bonding layer (not shown) such as a resin.Typically, a bonding layer will have a finite thickness and anassociated index of refraction. If a bonding layer is used, it can betreated as an additional substrate Accordingly, equation (15) can beused to compensate for the thickness and index of refraction of thebonding layer, as well as for that of the substrate. The associatedconic constant is determined and optimized in the manner previousdescribed.

In another alternative exemplary embodiment, the substrate 702 cancomprise an adhesive layer.

In another alternative exemplary embodiment, the substrate 702 cancomprise an opaque substrate. For example, the opaque substrate cancomprise paper. Additionally, the interlaced image can be printed on afront surface 706 of the opaque substrate. Then, the opaque substratecan be laminated to the lenticular lens array 400. In that case, theimage is located at the rear surface 407 of the array 400. Accordingly,the thickness of the substrate does not have to be considered todetermine the proper thickness T. However, if a bonding layer is used tolaminate the opaque substrate to the array 400, then the thickness ofthe bonding layer should be considered to determine the proper thicknessT.

In another alternative exemplary embodiment, the lenticules 402-406 canbe cast onto the substrate 702 such that a discontinuity exists betweenone or more pairs of adjacent lenticules. For example, the lenticules402-406 can be cast onto the substrate 702 such that a discontinuityexists between lenticules 402 and 404 or between lenticules 404 and 406.

A tool 800 for producing an elliptically-shaped lens element accordingto an exemplary embodiment of the present invention will now bedescribed with reference to FIGS. 8A and B. Tool 800 can be constructedfrom diamond or other suitable material. FIG. 8A illustrates across-section of a base member 802 for producing the tool 800 forforming elliptically-shaped lens elements according to an exemplaryembodiment of the present invention. FIG. 8B illustrates a front view ofthe tool 800 for forming elliptically-shaped lens elements according toan exemplary embodiment of the present invention.

The tool 800 can be is used to produce a regular array of groves in amandrel for casting or extruding the lenticular lens array. The tool 800is not used to directly form the lenticular lens array. For example, themandrel can comprise a drum, and the tool 800 can produce a spiral orscrew pattern in the drum. Alternatively, the tool 800 can produce astraight-cut (parallel-grooved) pattern in the drum. Furthermore, themandrel can be coated with a copper alloy prior to being shaped by thetool 800. The copper alloy can be used because it cuts cleanly and holdsit shape. After cutting, the copper alloy can be plated with anothermaterial to improve: the mandrel's durability. For example, the platingmaterial can comprise chrome. If a coating or plating material is usedafter cutting, then the dimensions of the tool 800 can be adjusted(increased) to compensate for a finite thickness of the coating orplating material. The following description details a tool that createsa mandrel without a coating or plating. In practice, the size of thetool 800 can account for the added thickness of the coating or plating.

As shown in the exemplary embodiments of FIGS. 8A and 8B, the basemember 802 can comprise a cylindrical rod and can have a radius bcorresponding to the dimensions ±b of an elliptical shape 806 for thetool 800. The base member 802 can be cut along a plane 804 at an angle kto a minor axis 808 of the base member 802. The angle k can bedetermined from the following equation: $\begin{matrix}{{{cosine}(k)} = \frac{b}{a}} & (16)\end{matrix}$

The elements b and a correspond to elliptical characteristics of theelliptical shape 806. The elliptical shape 806 corresponds to thedesired elliptical shape of lenticular lens elements on a lenticularlens array according to an exemplary embodiment of the presentinvention. Accordingly, each of the elliptical characteristics a, b, andc, correspond to the same characteristics for the elliptically shapedlens elements of the array.

In an alternative exemplary embodiment, the base member 802 can comprisea cone. The cone can comprise a truncated cone. The cone can comprisediamond or other suitable material. Standard geometric equations can beused to determine a proper angle to cut the cone to produce the desiredelliptical shape for the tool 800. Accordingly, the cone can be cut atan angle to produce the desired elliptical shape for the tool. Potentialadvantages of a truncated conical base member 802 include less materialbeing required and the conical apex angle providing a general reductionin angular range capability of the fabrication equipment.

The tool 800 comprises a mother tool that can be used to cut a mandrelfor producing elliptically-shaped lens elements of the array. Themandrel then can be used to create the elliptically-shaped lens elementsin a lenticular lens array. For example, the mandrel can be used forcasting or extruding the lenticular lens elements of the array.

FIG. 9 illustrates a pseudo elliptical lenticule 900 for approximatingan elliptically shaped lens element of a lenticular lens array accordingto an exemplary embodiment of the present invention. The lenticule 900can be included in a lenticular lens array according to an exemplaryembodiment of the present invention. The lenticule 900 comprises apseudo elliptical lens element 901. As shown, the pseudo elliptical lenselement 901 approximates a portion of an elliptical shape 902. Thepseudo elliptical lens element 901 comprises a circular portion 905,corresponding straight portions 906 a, 906 b, and corresponding straightportions 908 a, 908 b.

The circular portion 905 comprises a portion of a circular shape 904that approximates the radius R of the elliptical shape 902. Accordingly,the circular shape 904 can have a radius equal to the radius R of theelliptical shape. Alternatively, the circular shape 904 can have aradius different from the radius R of the elliptical shape, if thedifferent radius can better approximate the elliptical shape. Thecircular portion 905 can comprise that portion of the circular shape 904that approximates the elliptical shape 902 within a specified tolerance.The specified tolerance can be determined based on a desired projectedimage quality for a particular application. The maximum residual shapeerror of the circular and straight regions can be maintained to beapproximately the same.

Corresponding straight portions 906 a, 906 b can be provided beginningat a point where the circular shape 904 exceeds the specified tolerancefrom the elliptical shape 902. Accordingly, straight portions 906 a, 906b can approximate a portion of the desired elliptical shape 902.

Corresponding straight portions 908 a, 908 b can be provided beginningat a point where the straight portions 906 a, 906 b, respectively,exceed the specified tolerance from the elliptical shape 902. Thestraight portions 908 a, 908 b can approximate a portion of the desiredelliptical shape 902.

Any number of straight portions can be used to approximate theelliptical shape 902. The number of straight portions can be adjusted tominimize deviation from the elliptical shape 902. For example, usingmore straight portions can achieve less deviation from the desiredelliptical shape 902. In other words, a smaller tolerance limit can beused when more straight portions are used. Typically, if more straightportions are used, then a smaller circular portion 905 can be used toallow a smaller tolerance limit.

In an alternative exemplary embodiment, a plurality of facets can beused to approximate the desired elliptical shape without using acircular portion. In one embodiment, corresponding pairs of facets canbe used to approximate the desired elliptical shape. In that embodiment,the pseudo elliptical lens elements can have a point where a facet pairmeets at the vertex of the lens element. In an alternative embodiment,the vertex can be approximated with a single facet positionedsubstantially orthogonal to the major axis of the elliptical shape, andcorresponding pairs of facets can be used to approximate outer portionsof the elliptical shape.

Accordingly, the pseudo elliptical lens element 901 can approximate anelliptical shape 902, thereby improving the image characteristics in asimilar manner as described above for the array 400 of FIG. 4.

FIG. 10 illustrates a pseudo elliptical tool 1000 for creating a pseudoelliptical lens element according to an exemplary embodiment of thepresent invention. The tool 1000 can be is used to produce a regulararray of groves in a mandrel for casting or extruding the lenticularlens array. The tool 1000 is not used to directly form the lenticularlens array. For example, the mandrel can comprise a drum, and the tool1000 can produce a spiral or screw pattern in the drum. Alternatively,the tool 1000 can produce a straight-cut (parallel grooved) pattern inthe drum. Furthermore, the mandrel can be coated with a copper alloyprior to being shaped by the tool 1000. The copper alloy can be usedbecause it cuts cleanly and holds it shape. After cutting, the copperalloy can be plated with another material to improve the mandrel'sdurability. For example, the plating material can comprise chrome. If acoating or plating material is used after cutting, then the dimensionsof the tool 1000 can be adjusted (increased) to compensate for a finitethickness of the coating or plating material. The following descriptiondetails a tool that creates a mandrel without a coating or plating. Inpractice, the size of the tool 1000 can account for the added thicknessof the coating or plating.

In FIG. 10, only one side of the elliptical tool 1000 is illustrated.The other side of the elliptical tool 1000 comprises a mirror image ofthe illustrated side. The elliptical tool 1000 can be constructed fromdiamond or other suitable material. The elliptical tool 1000 cancomprise a tool for cutting a form that can be used to extrude or castpseudo elliptical lens elements of a lenticular lens array according toan exemplary embodiment of the present invention.

As shown, the desired elliptical shape comprises an elliptical shape1002. A circular shape 1004 having a radius can approximate a portion1005 of the elliptical shape 1002. Point 1007 indicates an intersectionof a tangent 1006 to the ellipse 1004 where the circular shape 1004exceeds a specified tolerance from the elliptical shape 1002. A firstfacet 1008 can be provided beginning at the point 1007 and canapproximate a portion of the elliptical shape 1002. A second facet 1010can be provided beginning at a point 1009 where the first facet exceedsthe specified tolerance from the elliptical shape 1002. The second facet1010 can approximate a portion of the elliptical shape 1002 until thedesired width y is reached The tangent 1006, the first facet 1008, andthe second facet 1010 can form an angle l, m, and n, respectively, withthe major axis 1003 of the elliptical shape 1002.

The actual angles l, m, and n, the radius R, and the length of the firstand second facets 1008, 1010 can be determined for a particularapplication based on the characteristics of the elliptical shape 1002and the specified tolerance.

In practice, the radius of the circular shape 1004 can be chosen toapproximate a radius R of the elliptical shape 1002. Accordingly, thecircular shape 1004 can have a radius equal to the radius R of theelliptical shape. Alternatively, the circular shape 1004 can have aradius different from the radius R of the elliptical shape, if thedifferent radius can better approximate the elliptical shape. The chosenradius can be used until it exceeds the specified tolerance from theelliptical shape 1002. The angle m of the first facet 1008 can bedetermined based on the elliptical shape 1002 at the tangent point 1007.Similarly, the angle n of the second facet 1010 can be determined basedon the elliptical shape 1002 at the point 1009.

Any number of facets can be used to approximate the elliptical shape1002. The number of facets can be adjusted to minimize deviation fromthe elliptical shape 1002. For example, using more facets can achieveless deviation from the desired elliptical shape 1002. In other words, asmaller tolerance limit can be used when more facets are used.Typically, if more facets are used, then a smaller circular portion 1005can be used to allow a smaller tolerance limit.

The mitigation of the spherical aberration afforded by the inclusion ofthe elliptically shaped lens, when compared to conventional lenses,allows the utilization of thinner lenticular lenses to achieve the sameor better performance. A common metric used to express the lightgathering capability of a lens is know as the focal ratio or F-number(F/#). The focal ratio is simply defined at the ratio of the focallength of the lens divided by the diameter of the lens (specifically theentrance pupil of the lens). The spherical aberration of theconventional lens follows the well-known relationship of being directlyproportional to 1/(F/#)³. As a conventional lenticular lens is thinned(t becoming smaller) while maintaining the pitch, it is evident that theimage resolution/quality degrades quickly since the F# becomes smaller.For example, the resolution decreases by a factor of over 10 as the tchanges from 0.020 inch to 0.009 inch. Sheets of thinner lenticularlenses offer significant advantages when affixed to cylindrical objectsas explained elsewhere in this specification.

In an alternative exemplary embodiment, a plurality of facets can beused to approximate the desired elliptical shape without using acircular portion. In one embodiment, corresponding pairs of facets canbe used to approximate the desired elliptical shape. In that embodiment,the pseudo elliptical lens elements can have a point where a facet pairmeets at the vertex of the lens element. In an alternative embodiment,the vertex can be approximated with a single facet positionedsubstantially orthogonal to the major axis of the elliptical shape, andcorresponding pairs of facets can be used to approximate outer portionsof the elliptical shape.

The tool 1000 can be used to carve a mandrel having a pseudo ellipticalshape. The pseudo elliptical mandrel then can be used for casting orextruding lens elements having a pseudo elliptical shape for alenticular lens array.

The elliptical shape of the lenticular lens elements according to theexemplary embodiments of the present invention can provide the followingbenefits over conventional designs: producing less visible printprojected aberrations; providing higher printed image contrast;providing thinner gauge lenticular materials that maintain the printquality present in thicker gauge materials (for example, the thinnerlenticular materials can be produced with a thickness less than 0.012inch, and more specifically in the range of about 0.005 inch to about0.010 inch); providing print images with clearer and smaller serif typeand point sizes; providing the thinner lenticular material gauges thatcan be flexible enough to affix to cylindrical or truncated packagingcontainers, such as jars, bottles, beverage cups, cartons, etc. withoutde-laminating off the consumer packaging; providing the thinnerlenticular material gauges that can be adaptable to the packagingindustry's in-line labeling applicators for rotary roll fed blow down orwipe-down labeling systems; providing the thinner gauge materials thatcan reduce the thickness and weight per square inch of material, therebyreducing cost; providing increased lenticule viewing width area forbroader animated imaging techniques at a lower material thickness andfiner lens pitch; or reducing cross-talk and image ghosting.

The elliptical shape lens elements of the lenticular lens arrayaccording to the exemplary embodiments of the present invention can beused in the following printed product types and markets due to thethinner lenticular material possible with the elliptical design: entireouter lenticular packaging enhancements (box overwraps); segmentedapplied lenticular label coverage to outer packaging; pressuresensitive, non-pressure sensitive, self-adhesive, and non-self-adhesivelenticular label products; multi-ply, multi-substrate peel open pressuresensitive and non-pressure sensitive lenticular labels; lenticularlaminated to paperboard products; packaging in-packs and on packs;beverage cups having decorative partial or full lenticular cup wraps;video, dvd, or cd disc cover lenticular treatments; direct mail;magazine inserts; newspaper inserts; or contest and game sweepstakecomponents that comprise use of partial or full lenticular enhancements

Although specific embodiments of the present invention have beendescribed above in detail, the description is merely for purposes ofillustration. Various modifications of, and equivalent stepscorresponding to, the disclosed aspects of the exemplary embodiments, inaddition to those described above, can be made by those skilled in theart without departing from the spirit and scope of the present inventiondefined in the following claims, the scope of which is to be accordedthe broadest interpretation so as to encompass such modifications andequivalent structures.

What is claimed is:
 1. A method for producing a tool, the tool forcreating a mandrel for forming elliptically-shaped lens elements of alenticular lens array, comprising the steps of: providing a base memberhaving a radius b corresponding to a minor axis of an elliptical shape,the elliptical shape corresponding to a desired elliptical shape foreach lens element of the lenticular lens array; and cutting the basemember along a plane that forms an angle k with the minor axis of thedesired elliptical shape, wherein the elliptical shape comprises a majoraxis perpendicular to the minor axis, wherein a vertex of the desiredelliptical shape lies at a point a along the major axis, and wherein theangle k is given by the formula cosine(k)=b/a.
 2. The method accordingto claim 1, wherein the base member comprises a cylindrical shape. 3.The method according to claim 1, wherein the base member comprisesdiamond.
 4. The method according to claim 1, wherein the radius b andthe point a are adjusted from desired elliptical shape to compensate fora protective surface that will be on the mandrel after being created bythe tool.
 5. The method according to claim 1, wherein the base membercomprises a cone shape.